Though the homogeneous (source-free) Helmholtz equation can be separated in eleven coordinate systems, we require separability into transverse and longitudinal parts and that is possible only in Cartesian, cylindrical, parabolic cylindrical and elliptical cylindrical coordinates [1]}. The spatial evolution of the scalar potential \(\psi \)  can then be described by the transverse and the longitudinal parts; the transverse part \(\varphi (u_1, u_2)\)  will depend only on the transverse coordinates, \(u_1,u_2\) , and the longitudinal part \(Z(z)\)  will depend on the longitudinal coordinate \(z\) ; thus
\(\psi (u,v,z)=\varphi (u_1,u_2)Z(z).\) 
